square root problem that involves geometry

by kjhgyuio, Apr 5, 2025, 3:56 AM

If x is a nonnegative real number , find the minimum value of √x^2+4 + √x^2 -24x +153

geometry

algebra

surds

Inspired by bamboozled

by sqing, Apr 5, 2025, 2:35 AM

Let $a,b,c$ be reals such that $(a^2+1)(b^2+1)(c^2+1) = 27.$ Prove that $1-3\sqrt 3\leq ab + bc + ca\leq 6$

inequalities proposed

algebra

Range of ab + bc + ca

by bamboozled, Apr 5, 2025, 2:12 AM

Let $(a^2+1)(b^2+1)(c^2+1) = 9$ , where $a, b, c \in R$ , then the number of integers in the range of $ab + bc + ca$ is __

inequalities

Functional Equation

by AnhQuang_67, Apr 4, 2025, 4:50 PM

Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R}$

This post has been edited 1 time. Last edited by AnhQuang_67, 4 hours ago
Reason: oops my bad

function

algebra

functional equation

Regarding Maaths olympiad prepration

by omega2007, Apr 4, 2025, 3:13 PM

<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compiled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your perspective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.

This post has been edited 1 time. Last edited by omega2007, 4 hours ago
Reason: Spelling error

Tangent.

by steven_zhang123, Mar 23, 2025, 2:35 AM

In $\triangle ABC$ with $AB > BC$ , a tangent to the circumcircle of $\triangle ABC$ at point $B$ intersects the extension of $AC$ at point $D$ . $E$ is the midpoint of $BD$ , and $AE$ intersects the circumcircle of $\triangle ABC$ at $F$ . Prove that $\angle CBF = \angle BDF$ .

China TST

geometry

Assisted perpendicular chasing

by sarjinius, Mar 9, 2025, 3:41 PM

In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$ , let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$ . Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$ , and let $F$ be the point on line $AC$ such that $FA = FE$ . Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$ .
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.

Integer Coefficient Polynomial with order

by MNJ2357, Jan 12, 2019, 11:36 AM

Find all polynomials $P(x)$ with integer coefficients such that for all positive number $n$ and prime $p$ satisfying $p\nmid nP(n)$ , we have $ord_p(n)\ge ord_p(P(n))$ .

This post has been edited 1 time. Last edited by MNJ2357, Jan 12, 2019, 10:17 PM

algebra

polynomial

number theory

inquequality

by ngocthi0101, Sep 26, 2014, 1:18 AM

let $a,b,c > 0$ prove that
$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$

inequalities

inequalities unsolved

inequalities proposed

IMO ShortList 1998, algebra problem 1

by orl, Oct 22, 2004, 2:46 PM

Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$ . Prove that

$\frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}.$